what is the difference between a damper motor and blower motor and their functions?
The blower motor moves air around you house. Runs with both heat and air conditioning. The "damper" motor (sometimes called inducers) helps moves air up the chimney so that dangerous exhaust gases don't enter the house. Furnances with damper motors usually have sensors to ensure it's functioning properly before the burner will light.
I don't know anything about mobile homes but I'd suspect the draft motor helps move combusion air in and out of the furnance. Combusion air is the air used to burn the fuel (natural gas, propane) and after the fuel burns contains harmful compounds like carbon monoxide. I'd also suspect that there is a sensor of some sort that ensures that blower in functioning properly before the burners will light.
As for troubleshooting, that would very much depend on your skill level and the tools you have available (or are willing to purchase). Furnances aren't the most complicated things in the world but do require an understanding of basic electrical testing and mechanical skills to disassemble the unit partly.